Verify the identity. $$ \sin ^{2} \frac{ heta}{2}=\frac{\sec heta-1}{2 \sec heta} $$ Use the appropriate power-reducing formula and rewrite the left side of the identity. $$ \frac{1-\cos heta}{2} $$ (Simplify your answer.) Rewrite the expression from the previous step by multiplying the numerator and denominator by sec $$ heta $$. Multiply and distribute in the numerator. $$ \square $$ (Do not simplify.)

Question

Verify the identity. $$ \sin ^{2} \frac{	heta}{2}=\frac{\sec 	heta-1}{2 \sec 	heta} $$ Use the appropriate power-reducing formula and rewrite the left side of the identity. $$ \frac{1-\cos 	heta}{2} $$ (Simplify your answer.) Rewrite the expression from the previous step by multiplying the numerator and denominator by sec $$ 	heta $$. Multiply and distribute in the numerator. $$ \square $$ (Do not simplify.)

UpStudy AI · Accepted Answer

To verify the identity

$$
\sin^2 \frac{\theta}{2} = \frac{\sec \theta - 1}{2 \sec \theta},
$$

we start with the left side and use the power-reducing formula for $$\sin^2 \frac{\theta}{2}$$:

$$
\sin^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{2}.
$$

Now, we rewrite the left side:

$$
\sin^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{2}.
$$

Next, we will rewrite this expression by multiplying the numerator and denominator by $$\sec \theta$$:

$$
\frac{1 - \cos \theta}{2} \cdot \frac{\sec \theta}{\sec \theta} = \frac{(1 - \cos \theta) \sec \theta}{2 \sec \theta}.
$$

Now, we distribute $$\sec \theta$$ in the numerator:

$$
(1 - \cos \theta) \sec \theta = \sec \theta - \cos \theta \sec \theta.
$$

Thus, we have:

$$
\frac{(1 - \cos \theta) \sec \theta}{2} = \frac{\sec \theta - \cos \theta \sec \theta}{2}.
$$

So, the expression we have after multiplying and distributing is:

$$
\frac{\sec \theta - \cos \theta \sec \theta}{2}.
$$

This is the expression we were looking for, and we can see that it matches the right side of the original identity:

$$
\frac{\sec \theta - 1}{2 \sec \theta}.
$$

Thus, we have verified the identity. The final answer for the expression after multiplying and distributing is:

$$
\frac{\sec \theta - \cos \theta \sec \theta}{2}.
$$
$$\frac{\sec \theta - \cos \theta \sec \theta}{2}$$

Verify the identity.

Use the appropriate power-reducing formula and rewrite the left side of the identity.
(Simplify your answer.)
Rewrite the expression from the previous step by multiplying the numerator and denominator by sec . Multiply and distribute in the numerator.
(Do not simplify.)

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Verify the identity. Use the appropriate power-reducing formula and rewrite the left side of the identity. (Simplify your answer.) Rewrite the expression from the previous step by multiplying the numerator and denominator by sec . Multiply and distribute in the numerator. (Do not simplify.)